Method allowing to obtain an optimum model of a physical characteristic in a heterogeneous medium such as the subsoil

Abstract

A method for obtaining, by means of an inversion process, an optimum model of a physical characteristic in a heterogeneous medium (the impedance of an underground zone in relation to waves transmitted in the ground for example), by taking as the starting point an a priori model of the physical characterized that is optimized by minimizing a cost function dependent on differences between the optimized model which is sought and the known data, considering the a priori model. Construction of the a priori model comprises correlation by kriging between values of the physical quantity known at different points of the medium along discontinuities (strata directions). Uncertainties about the values of the physical quantity in the a priori model in relation to the corresponding values in the medium follow a covariance model that controls the inversion parameters more quantitatively. The characteristics of the covariance model are defined in connection with the structure of the data observed or measured in the medium. An application of the optimum model is location of hydrocarbon reservoirs.

What is claimed is:  
     
       1. A method for obtaining an optimized model representing an image of a distribution in a subterranean stratified heterogeneous medium of a physical quantity with respect to waves propagating in the medium, comprising: 
       obtaining data representing values of the physical quantity by means of measurements, recordings or observations at different points of the medium along strata directions;  
       forming an a priori model by kriging with a stationary covariance between the obtained data representing values of the physical quantity known at points of the medium along the strata directions with uncertainties about the values of the physical quantity in the a priori model at any point along the strata directions being described by a covariance including the stationary covariance that depends only on a distance vector between the points and a non-stationary term depending on a position of the points and on the respective distances between the points;  
       controlling inversion parameters from characteristics of the covariance including the stationary covariance; and  
       constructing by inversion the optimized model by minimizing a cost function depending on differences between the model being sought and the known data considering the a priori model.  
     
     
       2. A method in accordance with  claim 1 , wherein: 
       the physical quantity is impedance of the medium to propagation of the waves.  
     
     
       3. A method in accordance with  claim 1 , wherein: 
       the optimized model is used to locate hydrocarbon reservoirs.  
     
     
       4. A method in accordance with  claim 3 , wherein: 
       the data comprise seismic data and the medium contains at least one well crossing the medium.  
     
     
       5. A method as claimed in  claim 4 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       6. A method as claimed in  claim 5 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       7. A method as claimed in  claim 5 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       8. A method as claimed in  claim 4 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       9. A method as claimed in  claim 4 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       10. A method as claimed in  claim 3 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       11. A method as claimed in  claim 10 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       12. A method as claimed in  claim 10 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       13. A method as claimed in  claim 3 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       14. A method as claimed in  claim 3 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       15. A method in accordance with  claim 1 , wherein: 
       the data comprise seismic data and the medium contains at least one well crossing the medium.  
     
     
       16. A method as claimed in  claim 9 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       17. A method as claimed in  claim 16 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       18. A method as claimed in  claim 16 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       19. A method as claimed in  claim 15 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       20. A method as claimed in  claim 15 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       21. A method as claimed in  claim 1 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       22. A method as claimed in  claim 21 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       23. A method as claimed in  claim 21 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       24. A method as claimed in  claim 1 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       25. A method as claimed in  claim 1 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       26. A method in accordance with  claim 2 , wherein: 
       the optimized model is used to locate hydrocarbon reservoirs.  
     
     
       27. A method in accordance with  claim 26 , wherein: 
       the data comprise seismic data and the medium contains at least one well crossing the medium.  
     
     
       28. A method as claimed in  claim 27 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       29. A method as claimed in  claim 28 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       30. A method as claimed in  claim 28 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       31. A method as claimed in  claim 27 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       32. A method as claimed in  claim 27 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       33. A method as claimed in  claim 26 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       34. A method as claimed in  claim 33 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       35. A method as claimed in  claim 33 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       36. A method as claimed in  claim 26 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       37. A method as claimed in  claim 26 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       38. A method in accordance with  claim 2 , wherein: 
       the data comprise seismic data and the medium contains at least one well crossing the medium.  
     
     
       39. A method as claimed in  claim 38 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       40. A method as claimed in  claim 39 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       41. A method as claimed in  claim 38 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       42. A method as claimed in  claim 38 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       43. A method as claimed in  claim 38 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       44. A method as claimed in  claim 2 , comprising defining the inversion parameters to be compatible with an error covariance defined by: 
       
         
             C   ε ( x, y )= C   z ( {right arrow over (h)} )− t β( x ) K β( x+{right arrow over (h)} )  
         
       
       where (x,y) are any two points of the medium at a distance from one another, K is a kriging matrix, β(x) and β(x+{right arrow over (h)}) are kriging weights respectively at points x and y and {right arrow over (h)} is the distance vector between the two points. 
     
     
       45. A method as claimed in  claim 44 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       46. A method as claimed in  claim 44 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       47. A method as claimed in  claim 2 , comprising determining a mean covariance that is adjusted to a stationary exponential covariance model in order to define the inversion parameters. 
     
     
       48. A method as claimed in  claim 2 , comprising, at all the points, adjusting the covariance including the stationary covariance to a stationary exponential model in order to define local values of the inversion parameters. 
     
     
       49. A method for obtaining an optimized model representing an image of a distribution in a subterranean stratified heterogeneous medium of a physical quantity with respect to waves propagating in the medium, comprising: 
       obtaining data representing values of the physical quantity by means of measurements, recordings or observations at different points of the medium along strata directions;  
       forming an a priori model by kriging with a stationary covariance between the obtained data representing values of the physical quantity known at points of the medium along the strata directions with uncertainties about the values of the physical quantity in the a priori model at any point along the strata directions being described by a covariance including the stationary covariance that depends only on a distance vector between the points and a non-stationary term depending on a position of the points and on the respective distances between the points;  
       controlling inversion parameters from characteristics of the covariance including the stationary covariance; and  
       constructing by inversion the optimized model by iteratively minimizing a cost function depending on differences between the model being sought and the known data considering the a priori model.